Dual canonical bases and cluster algebras

Abstract: One of Fomin and Zelevinsky’s main motivations for cluster algebras was to study the dual canonical bases. Correspondingly, it had been long conjectured that the quantum cluster monomials (certain monomials of generators) belong to the dual canonical bases up to scalar multiples. In a geometric framework for cluster algebras, Fock and Goncharov expected that cluster algebras possess bases with good tropical properties. In this talk, we consider a large class of quantum cluster algebras called injective-reachable (equivalently, there exists a green to red sequence). We study their tropical properties. Then we introduce the (common) triangular basis, which is a Kazhdan-Lusztig type basis with good tropical properties. We verify the above motivational conjecture in full generality.

Mathematics

Audience: researchers in the topic

ICCM 2020